Variational phase field model for dynamic brittle fracture

Transcription

1 Variational phase field model for dynamic brittle fracture Bleyer J., Roux-Langlois C., Molinari J-F. EMMC 15, September 8th, / 18

2 Outline Mechanisms of dynamic fracture Variational phase-field model of brittle fracture Crack branching in homogeneous medium Crack propagation in heterogeneous medium 1 / 18

3 Crack velocity Limiting crack velocity : in theory, v lim = c R for mode I never attained in experiments, rarely exceed c R seems to depend on experimental setup (geometry, loading conditions) 2 / 18

4 Crack velocity Limiting crack velocity : in theory, v lim = c R for mode I never attained in experiments, rarely exceed c R seems to depend on experimental setup (geometry, loading conditions) explained by crack tip instabilities [Sharon and Fineberg, 1996]: microbranching ( 0.4c R ) : small (1-100 µm in PMMA) short-lived micro-cracks, highly localized mirror, mist, hackle patterns 2 / 18

5 Crack branching Macroscopic branching at even higher velocities [Ramulu and Kobayashi, 1984] [Kobayashi and Mall, 1977] 3 / 18

6 Crack branching Macroscopic branching at even higher velocities [Ramulu and Kobayashi, 1984] [Kobayashi and Mall, 1977] Criterion for branching? question is still open... experiments and numerical simulations seem to exclude a criterion based (only) on crack tip velocity existence of a critical SIF or ERR? 3 / 18

7 Velocity-toughening mechanism Experiments on PMMA report a strong increase of apparent fracture energy with velocity : velocity-toughening mechanism a large part is attributed to an increase of created fracture surface due to microbranching recent experiments show an increase from 400 J/m 2 to J/m 2 between 0.11c R and 0.18c R [Scheibert et al., 2010] 4 / 18

8 Outline Mechanisms of dynamic fracture Variational phase-field model of brittle fracture Crack branching in homogeneous medium Crack propagation in heterogeneous medium 4 / 18

9 Phase-field approach alternative to cohesive elements or XFEM for simulating crack propagation non-local approach : continuous scalar field d(x) representing the crack + a regularization length l 0 [Bourdin et al., 2000] can be formulated as a damage gradient model 5 / 18

10 Phase-field approach alternative to cohesive elements or XFEM for simulating crack propagation non-local approach : continuous scalar field d(x) representing the crack + a regularization length l 0 [Bourdin et al., 2000] can be formulated as a damage gradient model convergence to Griffith theory when l 0 /L 0, at least for quasi-static propagation 5 / 18

11 Phase-field approach Many constitutive modeling choices are possible, we follow [Li et al., 2016] elastic strain energy density : ψ(ε, d) = (1 d) 2 ( κ 2 tr ε + + µε d : ε d) + κ 2 tr ε 6 / 18

12 Phase-field approach Many constitutive modeling choices are possible, we follow [Li et al., 2016] elastic strain energy density : ψ(ε, d) = (1 d) 2 ( κ 2 tr ε + + µε d : ε d) + κ 2 tr ε non-local fracture energy : w frac (d, d) = 3G c 8l 0 ( d + l 2 0 d 2) Remark : existence of an elastic phase for this model 6 / 18

13 Phase-field approach Many constitutive modeling choices are possible, we follow [Li et al., 2016] elastic strain energy density : ψ(ε, d) = (1 d) 2 ( κ 2 tr ε + + µε d : ε d) + κ 2 tr ε non-local fracture energy : w frac (d, d) = 3G c 8l 0 ( d + l 2 0 d 2) Remark : existence of an elastic phase for this model Numerical resolution using a staggered approach : minimization of total energy with respect to u : explicit dynamics minimization with respect to d : quadratic function with bound constraints (d n d n+1 1) to enforce damage irreversibility 6 / 18

14 Outline Mechanisms of dynamic fracture Variational phase-field model of brittle fracture Crack branching in homogeneous medium Crack propagation in heterogeneous medium 6 / 18

15 Prestrained plate geometry Prestrained PMMA plate, fixed boundaries [Zhou, 1996] E = 3.09 GPa, ν = 0.35, ρ = 1180 kg/m 3, G c = 300 J/m 2, c R = 906 m/s 7 / 18

16 Prestrained plate geometry Prestrained PMMA plate, fixed boundaries [Zhou, 1996] E = 3.09 GPa, ν = 0.35, ρ = 1180 kg/m 3, G c = 300 J/m 2, c R = 906 m/s strip geometry Γ = 2E( U) 2 /h crack should accelerate to c R transition from straight propagation to branched patterns apparent toughness increases with loading/crack velocity 7 / 18

17 Prestrained plate geometry (a) U = mm at t = 40 µs (b) U = mm at t = 40 µs (c) U = mm at t = 40 µs (d) U = mm at t = 20 µs 8 / 18

18 Prestrained plate geometry (a) U = mm at t = 40 µs (b) U = mm at t = 40 µs (c) U = mm at t = 40 µs (d) U = mm at t = 20 µs however : branching occurs at smaller load levels than in experiments, crack is too fast same problem with CZM, non-local integral approach 8 / 18

19 Crack velocities no evident decrease of crack speed after branching limiting velocity around 0.68c R 9 / 18

20 Damage zone thickening 10 / 18

21 Damage zone thickening 10 / 18

22 Damage zone thickening 10 / 18

23 Damage zone thickening 10 / 18

24 Damage zone thickening 10 / 18

25 Damage zone thickening 10 / 18

26 Damage zone thickening 10 / 18

27 Damage zone thickening progressive thickening of the damaged band before branching similar observation using peridynamics branching viewed as a progressive transition from a widening crack to two crack tips screening each other branching angle seems to depend on geometry 10 / 18

28 Apparent fracture energy Damage dissipation rate Γ = de frac /da interpreted as the apparent fracture energy 11 / 18

29 Apparent fracture energy Damage dissipation rate Γ = de frac /da interpreted as the apparent fracture energy 11 / 18

30 Apparent fracture energy Damage dissipation rate Γ = de frac /da interpreted as the apparent fracture energy 11 / 18

31 Apparent fracture energy Damage dissipation rate Γ = de frac /da interpreted as the apparent fracture energy suggests a critical value of Γ 2G c associated to branching 11 / 18

32 Velocity-toughening mechanism during propagation and before macroscopic branching 12 / 18

33 Velocity-toughening mechanism during propagation and before macroscopic branching existence of a well-defined Γ(v) relationship associated to a velocity-toughening mechanism 12 / 18

34 Velocity-toughening mechanism the Γ(v) relationship seems material-independent but geometry-dependent 13 / 18

35 Outline Mechanisms of dynamic fracture Variational phase-field model of brittle fracture Crack branching in homogeneous medium Crack propagation in heterogeneous medium 13 / 18

36 Propagation in constrained path experiments report that crack can reach c R if constrained in a weak plane [Washabaugh and Knauss, 1994] Loading U (mm) Stored energy (N/m) Crack velocity (c R ) , , / 18

37 Propagation in constrained path idem for a series of holes on crack path D = 0.4 mm and S = 0.9 mm 15 / 18

38 Propagation in constrained path idem for a series of holes on crack path 15 / 18

39 Propagation in constrained path idem for a series of holes on crack path velocity of 0.9c R for U = 0.05 mm shares qualitative similarities the nucleation and growth of microcracks interacting with defects the apparent fracture energy is much higher than the average toughness G c,weak = (1 D/S)G c 0.56G c 15 / 18

40 Interaction with distant heterogeneities crack passing near a hole 1mm from notch 16 / 18

41 Interaction with distant heterogeneities crack passing near a hole 1mm from notch 6mm from notch 16 / 18

42 Interaction with distant heterogeneities crack passing near a hole 1mm from notch 6mm from notch velocity of the crack tip is larger in the second case crack is more attracted : different near-tip stress fields? faster crack looks for other ways of dissipating energy? 16 / 18

43 Interaction with out-of-plane heterogeneities Configuration with an array of holes located away from the middle plane B = 0.5 mm offset, U = 0.04 mm B = 0.5 mm offset, U = 0.05 mm B = 0.6 mm offset, U = 0.04 mm B = 0.6 mm offset, U = 0.05 mm 17 / 18

44 Conclusions and perspectives Conclusion : some physical aspects of dynamic fracture can be reproduced with the phase-field approach propagation characterized by a damage band widening widening associated to an increase of the apparent fracture energy existence of a well-defined Γ(v) relationship macroscopic branching observed when Γ 2G c existence of a limiting velocity around 0.7c R c R can be reached in constrained geometries strong influence of heterogeneities on branching process 18 / 18

45 Conclusions and perspectives Conclusion : some physical aspects of dynamic fracture can be reproduced with the phase-field approach propagation characterized by a damage band widening widening associated to an increase of the apparent fracture energy existence of a well-defined Γ(v) relationship macroscopic branching observed when Γ 2G c existence of a limiting velocity around 0.7c R c R can be reached in constrained geometries strong influence of heterogeneities on branching process Open questions rate-dependent model for PMMA? energy-based branching criterion? better understanding of 3D effects and role of defects 18 / 18